Week 15: Code along 13
CHAPTER 11 Monte Carlo Simulation
Tyler Roden
2024-12-02
# Load packages
# Core
library(tidyverse)
library(tidyquant)
library(dplyr)
# time series
library(timetk)Goal
Simulate future portfolio returns
five stocks: “SPY”, “EFA”, “IJS”, “EEM”, “AGG”
market: “SPY”
from 2012-12-31 to 2017-12-31
1 Import stock prices
symbols <- c("SPY", "EFA", "IJS", "EEM", "AGG")
prices <- tq_get(x = symbols,
get = "stock.prices",
from = "2012-12-31",
to = "2017-12-31")2 Convert prices to returns
asset_returns_tbl <- prices %>%
group_by(symbol) %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
type = "log") %>%
slice(-1) %>%
ungroup() %>%
set_names(c("asset", "date", "returns"))3 Assign a weight to each asset
# symbols
symbols <- asset_returns_tbl %>% distinct(asset) %>% pull()
symbols## [1] "AGG" "EEM" "EFA" "IJS" "SPY"# weights
weights <- c(0.25, 0.25, 0.2, 0.2, 0.1)
weights## [1] 0.25 0.25 0.20 0.20 0.10w_tbl <- tibble(symbols, weights)
w_tbl## # A tibble: 5 × 2
## symbols weights
## <chr> <dbl>
## 1 AGG 0.25
## 2 EEM 0.25
## 3 EFA 0.2
## 4 IJS 0.2
## 5 SPY 0.14 Build a portfolio
# ?tq_portfolio
portfolio_returns_tbl <- asset_returns_tbl %>%
tq_portfolio(assets_col = asset,
returns_col = returns,
weights = w_tbl,
rebalance_on = "months",
col_rename = "returns")
portfolio_returns_tbl## # A tibble: 60 × 2
## date returns
## <date> <dbl>
## 1 2013-01-31 0.0204
## 2 2013-02-28 -0.00239
## 3 2013-03-28 0.0121
## 4 2013-04-30 0.0174
## 5 2013-05-31 -0.0128
## 6 2013-06-28 -0.0247
## 7 2013-07-31 0.0321
## 8 2013-08-30 -0.0224
## 9 2013-09-30 0.0511
## 10 2013-10-31 0.0301
## # ℹ 50 more rows5 Simulating growth of a dollar
# Get mean portfolio return
mean_port_return <- mean(portfolio_returns_tbl$returns)
mean_port_return## [1] 0.005899138# Get standard deviation of portfolio returns
stddev_port_return <- sd(portfolio_returns_tbl$returns)
stddev_port_return## [1] 0.02347492# Construct a normal distribution
simulated_monthly_returns <- rnorm(120, mean_port_return, stddev_port_return)
simulated_monthly_returns## [1] -0.0054444156 -0.0090687643 -0.0089247806 -0.0020755011 -0.0071155426
## [6] 0.0242941588 0.0074297062 0.0136965180 -0.0083679715 -0.0123705328
## [11] -0.0260312456 0.0070056777 -0.0503035822 0.0412592985 0.0229225260
## [16] 0.0202927081 0.0122888792 0.0483434339 0.0263589389 0.0145232002
## [21] 0.0448416424 0.0008030043 0.0294142895 0.0104578015 -0.0089182025
## [26] 0.0104112854 0.0091893880 -0.0023382669 -0.0072108427 -0.0089403113
## [31] -0.0119553452 -0.0112018659 -0.0036278691 0.0315472770 -0.0378044761
## [36] 0.0331140395 0.0081488033 0.0028361492 0.0229176185 0.0341535751
## [41] 0.0021324813 -0.0098557549 -0.0192127204 0.0177601981 -0.0099561577
## [46] 0.0181680497 -0.0229725296 0.0293464594 -0.0214801677 -0.0063529242
## [51] -0.0017945144 -0.0071839433 -0.0177100371 0.0175903446 0.0302137506
## [56] -0.0073072818 -0.0332220058 -0.0164796839 0.0639892675 0.0458763969
## [61] -0.0111167956 0.0079005021 0.0008123683 -0.0026268755 0.0318621582
## [66] 0.0103503608 0.0162813198 0.0142797604 0.0281737925 0.0664998842
## [71] 0.0413230566 0.0039221272 -0.0077149617 -0.0067063465 0.0193404348
## [76] -0.0174274878 0.0187139044 -0.0132086273 0.0038155884 0.0342164284
## [81] 0.0321210599 -0.0036699759 0.0448444069 0.0200938029 0.0026413822
## [86] 0.0002789540 0.0220986526 0.0038441254 -0.0230928312 0.0472171090
## [91] 0.0047404182 -0.0400010194 -0.0071099847 0.0289139465 -0.0178653612
## [96] 0.0145137685 0.0161117735 0.0104330511 0.0006858834 0.0077759681
## [101] 0.0230238198 -0.0065053082 0.0190082118 -0.0407711044 0.0231503386
## [106] 0.0198932428 -0.0152585975 0.0237599320 0.0239282826 0.0275671442
## [111] 0.0241541192 -0.0154011381 -0.0046715137 -0.0198577893 -0.0176805732
## [116] -0.0132622863 0.0199138326 0.0287938368 -0.0029116871 -0.0121797685# Add a dollar
simulated_returns_add_1 <- tibble(returns = c(1, 1 + simulated_monthly_returns))
simulated_returns_add_1## # A tibble: 121 × 1
## returns
## <dbl>
## 1 1
## 2 0.995
## 3 0.991
## 4 0.991
## 5 0.998
## 6 0.993
## 7 1.02
## 8 1.01
## 9 1.01
## 10 0.992
## # ℹ 111 more rows# Calculate the cumulative growth of a dollar
simulated_growth <- simulated_returns_add_1 %>%
mutate(growth = accumulate(returns, function(x, y) x*y)) %>%
select(growth)
simulated_growth## # A tibble: 121 × 1
## growth
## <dbl>
## 1 1
## 2 0.995
## 3 0.986
## 4 0.977
## 5 0.975
## 6 0.968
## 7 0.991
## 8 0.999
## 9 1.01
## 10 1.00
## # ℹ 111 more rows# Check the compound annual growth rate
cagr <- ((simulated_growth$growth[nrow(simulated_growth)]^(1/10)) - 1) * 100
cagr## [1] 7.9334556 Simulation function
simulate_accumulation <- function(initial_value, N, mean_return, sd_return) {
# Add a dollar
simulated_returns_add_1 <- tibble(returns = c(initial_value, 1 + rnorm(N, mean_return, sd_return)))
# Calculate the cumulative growth of a dollar
simulated_growth <- simulated_returns_add_1 %>%
mutate(growth = accumulate(returns, function(x, y) x*y)) %>%
select(growth)
return(simulated_growth)
}
simulate_accumulation(initial_value = 100, N = 240, mean_return = 0.005, sd_return = .01) %>%
tail()## # A tibble: 6 × 1
## growth
## <dbl>
## 1 380.
## 2 386.
## 3 394.
## 4 396.
## 5 399.
## 6 402.dump(list = c("simulate_accumulation"),
file = "../00_scripts/simulate-accumulation.R")7 Running multiple simulations
# Create a vector of 1s as a starting point
sims <- 51
starts <- rep(1, sims) %>%
set_names(paste0("sim", 1:sims))
starts## sim1 sim2 sim3 sim4 sim5 sim6 sim7 sim8 sim9 sim10 sim11 sim12 sim13
## 1 1 1 1 1 1 1 1 1 1 1 1 1
## sim14 sim15 sim16 sim17 sim18 sim19 sim20 sim21 sim22 sim23 sim24 sim25 sim26
## 1 1 1 1 1 1 1 1 1 1 1 1 1
## sim27 sim28 sim29 sim30 sim31 sim32 sim33 sim34 sim35 sim36 sim37 sim38 sim39
## 1 1 1 1 1 1 1 1 1 1 1 1 1
## sim40 sim41 sim42 sim43 sim44 sim45 sim46 sim47 sim48 sim49 sim50 sim51
## 1 1 1 1 1 1 1 1 1 1 1 1# Simulate
monte_carlo_sim_51 <- starts %>%
# Simulate
map_dfc(.x = .,
.f = ~simulate_accumulation(initial_value = .x,
N = 120,
mean_return = mean_port_return,
sd_return = stddev_port_return)) %>%
# Add column month
mutate(month = 1:nrow(.)) %>%
select(month, everything()) %>%
# Rearrange column names
set_names(c("month", names(starts))) %>%
# Transform to long form
pivot_longer(cols = -month, names_to = "sim", values_to = "growth")
monte_carlo_sim_51## # A tibble: 6,171 × 3
## month sim growth
## <int> <chr> <dbl>
## 1 1 sim1 1
## 2 1 sim2 1
## 3 1 sim3 1
## 4 1 sim4 1
## 5 1 sim5 1
## 6 1 sim6 1
## 7 1 sim7 1
## 8 1 sim8 1
## 9 1 sim9 1
## 10 1 sim10 1
## # ℹ 6,161 more rows# find quantiles
monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
pull(growth) %>%
quantile(probs = c(0, 0.25, 0.5, 0.75, 1)) %>%
round(2)## 0% 25% 50% 75% 100%
## 1.35 1.63 1.97 2.41 4.098 Visualizing simulations with ggplot
monte_carlo_sim_51 %>%
ggplot(aes(x = month, y = growth, color = sim)) +
geom_line() +
theme(legend.position = "none") +
theme(plot.title = element_text(hjust = 0.5)) +
labs(title = "Simulating growth of $1 over 120 months") 
Line plot with max, median, and min
# Step 1 Summarize data into max, median, and min of last value
sim_summary <- monte_carlo_sim_51 %>%
group_by(sim) %>%
summarise(growth = last(growth)) %>%
ungroup() %>%
summarise(max = max(growth),
median = median(growth),
min = min(growth))
sim_summary## # A tibble: 1 × 3
## max median min
## <dbl> <dbl> <dbl>
## 1 4.09 1.97 1.35# Step 2 Plot
monte_carlo_sim_51 %>%
# Filter for max, median, and min sim
group_by(sim) %>%
filter(last(growth) == sim_summary$max |
last(growth) == sim_summary$median |
last(growth) == sim_summary$min) %>%
ungroup() %>%
#plot
ggplot(aes(x = month, y = growth, color = sim)) +
geom_line() +
theme(legend.position = "none") +
theme(plot.title = element_text(hjust = 0.5)) +
theme(plot.subtitle = element_text(hjust = 0.5)) +
labs(title = "Simulating growth of $1 over 120 months",
subtitle = "Maximum, Median, and Minimum Simulation") 